Homework 5

Directions:

  1. Show each step of your work and fully simplify each expression.
  2. Turn in your answers in class on a physical piece of paper.
  3. Staple multiple sheets together.
  4. Feel free to use Desmos for graphing.

Answer the following:

  1. Add, subtract, divide or multiply. Make sure your answer is in the form $a + bi$.
    1. $\dfrac{1}{i}$ (hint: multiply by $\frac{i}{i}$).
    2. $\dfrac{2 - 3i}{1 - 2i}$
    3. $i^{10}$
    4. $i^{13}$
    5. $-i^{73}$
    6. $(-i)^{73}$
  2. Solve $3x^2 + 1 = 0$. Make sure your solution is a complex number.
  3. Draw a coordinate plane and graph the points $(1,2), (2, -1), (-3, -4)$ and $(4, -3)$.
  4. Suppose I have an expression $f(x)$ and I find two different inputs that give the same evaluation. In particular, I find that $x = 2$ and $x = 3$ gives $f(2) = f(3)$. Is $f$ a function?
  5. Suppose I have an expression $f(x)$ and I find one input which gives two different evaluations. In particular, I find that $x = 2$ spits out $f(2) = 5$ and $f(2) = 3$. Is $f$ a function?
  6. Write down two examples of functions in your daily life.
  7. Let $f(x) = x^2 - x + 1$. Evaluate and simplify the following:
    1. $f(1)$
    2. $f(a)$
    3. $f(-a)$
    4. $f(x + h)$
    5. $f(x + h) - f(x)$
  8. Let $f(x) = \dfrac{x + 1}{x^2 + x}$ Evaluate and simplify the following:
    1. $f(-1)$
    2. $f(x + 1)$
    3. $f(-x^2)$
  9. Let \[f(x) = \begin{cases} 3x + 2 & x \geq 2 \\ x^2 - x & x < 2\end{cases}\] Evaluate the following:
    1. $f(0)$
    2. $f(1)$
    3. $f(2)$
    4. $f(3)$
  10. Let \[f(x) = \begin{cases}-1 & x < -1 \\ 1 & -1 \leq x \leq 1 \\ -1 & x > 1\end{cases}\] Evaluate the following:
    1. $f(-2)$
    2. $f(-1)$
    3. $f(0)$
    4. $f(1)$
    5. $f(2)$
  11. Suppose $f$ is a function. What two problems do you need to look for when finding the domain?
  12. Find the domain of the following functions:
    1. $f(x) = \dfrac{1}{x - 1} + \dfrac{1}{x - 2}$
    2. $f(x) = \sqrt{2x + 1}$
    3. $f(x) = \dfrac{2}{3(x-1)\sqrt{2x + 3}} \ $ (hint: use the zero-product property on the denominator. factors???)
    4. $f(x) = \sqrt{x - 1} + \dfrac{1}{x - 3}$
    5. $f(x) = \dfrac{1}{x^2 + 3x + 2}$
  13. Sketch a graph on the coordinate plane for the following functions using the table method. Feel free to verify your graphs with Desmos.
    1. $f(x) = -x^2$
    2. $g(x) = x^3 + 1$
    3. $f(x) = \begin{cases} 1 & x \leq 1 \\ x + 1 & x > 1\end{cases}$
    4. $f(x) = \begin{cases} x & x > 1 \\ x^2 - 1 & x \leq -1\end{cases}$
    5. $f(x) = \begin{cases} 1 & x \geq 2 \\ -1 & x < 2 \end{cases}$
    6. $g(x) = \lvert x \rvert$
  14. (skip) NDraw a curve in the plane that is not the graph of a function.
  15. Common errors: Suppose your friend was given a function \[f(x) = x^2 - 3\] and they are asked to find $f(x + h) - f(x)$. Your friend ends up writing \[f(x + h) - f(x) = x^2 - 3 + h - x^2 - 3\] Identify the two mistakes that were made.